measure_le_lintegral_thickenedIndicator

English

With additional positivity of δ, the thickened indicator bounds the measure exactly via lintegral.

Русский

При 추가 положительности δ, толщиненный индикатор точно ограничивает меру через линеграл.

LaTeX

$$$$\\mu E \\le \\int^{\\! -}_{x} (thickenedIndicatorAux δ E x) \\, d\\mu$$$$

Lean4

theorem measure_le_lintegral_thickenedIndicator (μ : Measure X) {E : Set X} (E_mble : MeasurableSet E) {δ : ℝ}
    (δ_pos : 0 < δ) : μ E ≤ ∫⁻ x, (thickenedIndicator δ_pos E x : ℝ≥0∞) ∂μ :=
  by
  convert measure_le_lintegral_thickenedIndicatorAux μ E_mble δ
  dsimp
  simp only [thickenedIndicatorAux_lt_top.ne, ENNReal.coe_toNNReal, Ne, not_false_iff]

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